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Properties

Label 2-637-91.74-c1-0-35
Degree $2$
Conductor $637$
Sign $-0.927 - 0.374i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.414·2-s + (−0.707 − 1.22i)3-s − 1.82·4-s + (0.914 + 1.58i)5-s + (−0.292 − 0.507i)6-s − 1.58·8-s + (0.500 − 0.866i)9-s + (0.378 + 0.655i)10-s + (−0.292 − 0.507i)11-s + (1.29 + 2.23i)12-s + (−3.5 − 0.866i)13-s + (1.29 − 2.23i)15-s + 3·16-s − 5.82·17-s + (0.207 − 0.358i)18-s + (−3 + 5.19i)19-s + ⋯
L(s)  = 1  + 0.292·2-s + (−0.408 − 0.707i)3-s − 0.914·4-s + (0.408 + 0.708i)5-s + (−0.119 − 0.207i)6-s − 0.560·8-s + (0.166 − 0.288i)9-s + (0.119 + 0.207i)10-s + (−0.0883 − 0.152i)11-s + (0.373 + 0.646i)12-s + (−0.970 − 0.240i)13-s + (0.333 − 0.578i)15-s + 0.750·16-s − 1.41·17-s + (0.0488 − 0.0845i)18-s + (−0.688 + 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.927 - 0.374i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (3.5 + 0.866i)T \)
good2 \( 1 - 0.414T + 2T^{2} \)
3 \( 1 + (0.707 + 1.22i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.914 - 1.58i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.292 + 0.507i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 5.82T + 17T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.41T + 23T^{2} \)
29 \( 1 + (2.08 - 3.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.29 - 2.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.48T + 37T^{2} \)
41 \( 1 + (0.0857 - 0.148i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.70 + 2.95i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.82 - 3.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 + (2.08 - 3.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.12 + 3.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.82 - 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.32 + 9.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.878 + 1.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.07T + 83T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + (-5.41 - 9.37i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.16520675246298074068853173738, −9.255434022224386753122188716973, −8.352945097744837708475055744532, −7.21917998714636082607153865935, −6.45149239711639396395104514240, −5.63515935094153939837657519106, −4.53513858379358614263131347168, −3.39633099732029916415493261932, −1.95276472145636761828754934479, 0, 2.19592306652031372783635833388, 3.95720729443932931772682316162, 4.84823248232045327911259167677, 5.08562042895818766304510628573, 6.36846092663247169429493261788, 7.61228671327449205766684590908, 8.871708047891928019944508044385, 9.253008325006451091127163862066, 10.11612369182569308954140469769

Graph of the $Z$-function along the critical line